Method and device for hologram synthesis

ABSTRACT

The method consists of transforming a digital two-dimensional image defined by a real function into a complex two-dimensional image defined by a complex function, oversampling the complex image, simulating the production of a diffracted image resulting from the diffraction of an optical wave by the oversampled complex image, and adding a complex field representing a reference optical wave to the resulting diffracted image to produce a hologram. The hologram produced in this way can be used to produce images in three dimensions or in telecommunications.

The present invention relates generally to synthesizing holograms andmore particularly to synthesizing holograms digitally fromtwo-dimensional images stored in a memory.

Digital synthesis of holograms from two-dimensional images is used inmethods of reproducing three-dimensional images, for example. Respectiveholograms are computed for two-dimensional digital images representing athree-dimensional object from different viewpoints. These holograms arethen combined to produce a hologram of the object which reproduces athree-dimensional image of the object when it is reproduced physicallyby a spatial light modulator and illuminated by a coherent wave.

There are other applications of digital synthesis of holograms, inparticular in telecommunications, radar, X-rays and sonar.

Digital techniques for synthesizing holograms are known in the art. Forexample, the article by S. Michelin et al. entitled “Fourier-transformcomputer generated hologram: a variation on the off-axis principle”published in SPIE Conferences 1994, Practical Holography VIII, pages249-254, describes a method of simulating the production of an analoghologram. The method consists of applying a Fourier transform to atwo-dimensional image, adding a complex field representing a referenceoptical wave to the Fourier transform obtained in this way, and thenextracting the amplitude information contained in the sum of the complexfield and the Fourier transform. Applying the Fourier transform to thetwo-dimensional image digitally simulates the production of a“diffracted” image which results from the diffraction of a fictitiousoptical wave by the two-dimensional image. The two-dimensional image isalso oversampled before the Fourier transform is applied to it. However,the oversampled two-dimensional image obtained in this way is defined bya real intensity distribution which is not always well suited tocomputing a complex transform such as a Fourier transform.

The present invention aims to provide a method of synthesizing hologramsthat is more efficient than those of the prior art.

To this end, a method of producing a hologram from a two-dimensionalimage defined by a real function is characterized in that it comprisesthe following steps:

transforming the two-dimensional image defined by said real functioninto a complex two-dimensional image defined by a complex function,

oversampling the complex image,

simulating the production of a diffracted image resulting from thediffraction of an optical wave by the oversampled complex image, and

adding a complex field representing a reference optical wave to theresulting diffracted image in order to produce said hologram.

The method can further comprise the step of encoding values taken by theamplitude of the sum of said complex field and the resulting diffractedimage, so that the hologram can be reproduced on a liquid crystal screenor transmitted over a transmission line, for example.

In the present context, a “real or complex function” means a function oftwo variables, in the form of digital data, and taking real or complexvalues, respectively. The real function is typically an intensitydistribution while the complex function is a distribution of complexnumbers each defined by a real amplitude and a real phase.

The step of transforming the given two-dimensional image into a compleximage derives from the original two-dimensional image an image which isdefined by complex numbers which optimally represent the real opticalfield and facilitate the computations employed in the simulation step.

The oversampling step increases the number of pixels of the hologrambecause the computations employed in subsequent steps apply to a greaternumber of image points. This step can consist of inserting the compleximage into a larger image in which the intensity of pixels outside theoriginal complex image is made equal to 0. In this case, implementingthe step of oversampling the complex image after the steps oftransforming the two-dimensional image into a complex image avoidshaving to calculate the complex function for points of the oversampledimage outside the original complex image.

The transform step typically includes the following steps:

determining amplitude values each depending on the square root of acorresponding value taken by said real function, and

associating a phase with each of said amplitude values so that anamplitude value and a phase value are defined for each point of thecomplex image.

By averaging the amplitude values of the hologram, associating a phasewith each amplitude value avoids peaks of excessively high amplitude inthe resulting hologram of the given two-dimensional image.

The simulation step can include computing one of the following complextransforms: Fourier transform, Walsh transform, Hankel transform,orthogonal polynomial transform, Hadamar transform, Karhunen-Loevetransform, multiresolution discrete wavelet transform, adaptive wavelettransform and a transform which is a composite of at least two of theabove transforms.

The simulation step advantageously consists of computing a convolutionalproduct, associated with the oversampled complex image, of twocomponents, by applying the transform which is the inverse of saidcomplex transform to the product of the respective complex transforms ofsaid two components.

Until now, the skilled person has regarded the Fourier transform, whichis widely used in optics, as the best possible transform for calculatinga convolutional product of this kind. However, experiments conducted bythe present inventors have shown that using one of the complextransforms mentioned above other than the Fourier transform produces,for a two-dimensional image, a resultant hologram of much betterquality, i.e. which, when it is reproduced physically and illuminated bya coherent source, produces an image associated with the two-dimensionalimage that is finer than those generally produced by prior art systems.

According to another aspect of the invention, a method of producing ahologram from a two-dimensional image defined by a real function ischaracterized in that it comprises the following steps:

oversampling the two-dimensional image,

transforming the oversampled two-dimensional image into a complextwo-dimensional image defined by a complex function,

simulating the production of a diffracted image resulting from thediffraction of an optical wave by the oversampled complex image, and

adding a complex field representing a reference optical wave to theresulting diffracted image to produce said hologram.

The invention also provides a system for producing a hologram from atwo-dimensional image defined by a real function, characterized in thatit comprises:

transform means for transforming the two-dimensional image defined bysaid real function into a complex two-dimensional image defined by acomplex function,

means for oversampling the complex image,

simulator means for simulating the production of a diffracted imageresulting from the diffraction of an optical wave by the oversampledcomplex image, and

means for adding a complex field representing a reference optical waveto the resulting diffracted image to produce said hologram.

The system can further comprise means for encoding values taken by theamplitude of the sum of said complex field and the diffracted image.

The transform means can comprise means for determining amplitude valueseach depending on the square root of the corresponding value taken bysaid real function and means for associating a phase with each of saidamplitude values so that an amplitude value and a phase value aredefined for each point of the complex image.

The simulator means can comprise means for computing one of thefollowing complex transforms: Fourier transform, Walsh transform, Hankeltransform, orthogonal polynomial transform, Hadamar transform,Karhunen-Loeve transform, multiresolution discrete wavelet transform,adaptive wavelet transform and a transform which is a composite of atleast two of the above transforms.

The simulator means advantageously comprise means for computing aconvolutional product, associated with the oversampled complex image, oftwo components, by applying the transform which is the inverse of saidcomplex transform to the product of the respective complex transforms ofsaid two components.

According to another aspect of the invention, a system for producing ahologram from a two-dimensional image defined by a real function ischaracterized in that it comprises:

means for oversampling the two-dimensional image,

transform means for transforming the oversampled two-dimensional imageinto a complex two-dimensional image defined by a complex function,

simulator means for simulating the production of a diffracted imageresulting from the diffraction of an optical wave by the oversampledcomplex image, and

means for adding a complex field representing a reference optical waveto the resulting diffracted image to produce said hologram.

Other advantages of the present invention will become apparent onreading the following detailed description with reference to theaccompanying drawings, in which:

FIG. 1 is a flowchart of an algorithm according to the invention,

FIG. 2 is a block diagram of a computer executing the algorithm shown inFIG. 1,

FIG. 3 illustrates the production of a hologram from a two-dimensionalimage,

FIG. 4 shows the oversampling of a two-dimensional image by thealgorithm shown in FIG. 1, and

FIG. 5 is a diagram showing geometrical planes used in the algorithmshown in FIG. 1.

FIG. 1 shows an algorithm of the invention for synthesizing digitalholograms, which is executed by a microprocessor MP associated with amemory MM, both shown in FIG. 2.

In a preliminary step EO, a two-dimensional digital image IM, showndiagrammatically in FIG. 3, is stored in the memory MM associated withthe microprocessor MP in the form of digital data. The two-dimensionalimage IM is typically defined by a real function of two variables,particularly by a distribution of intensities f(x,y), where (x,y)represent co-ordinates in a two-dimensional system of axes (O,x,y)associated with the image IM.

In a step E1, the two-dimensional image IM is transformed into atransformed two-dimensional image IM1 which is defined by an amplitudedistribution by computing for each point of the image IM a valueproportional to the square root of the corresponding intensity value.

In the next step E2, a “pseudorandom” diffuser is generated digitally.This diffuser consists of an “image” having the same number of pixels asthe two-dimensional image IM and in which each pixel has an intensityvalue equal to 1 and a random phase. Each phase of the diffuser is thenassociated with a corresponding pixel of the transformed two-dimensionalimage IM1, to transform the image IM1 into a “complex” image IM2 inwhich a complex number defined by an amplitude value and a phase valueis determined for each pixel. The pseudorandom diffuser prevents theresulting hologram HO, shown diagrammatically in FIG. 3, associated withthe image IM, having excessive amplitude level disparities by averagingthe amplitude values of the hologram.

In a step E3 the complex image IM2 obtained in step E2 is oversampled,i.e. the image is included in a larger image, as shown in FIG. 4. Anoversampled image IM3 is formed in this way consisting of the compleximage IM2 in a central part PC and of pixels whose amplitude is chosenarbitrarily, for example equal to 0, in a complementary periphery partPF. This oversampling of the complex image IM2 increases the number ofpixels of the resultant hologram HO and therefore improves itsresolution.

In a step E4, the production of a diffracted image IM4 resulting fromthe diffraction of a fictitious coherent optical wave DIF by theoversampled complex image IM3 is simulated digitally. To this end, firstand second parallel and separate geometrical planes P1 and P2 aredefined within a three-dimensional system of axes (O′,X,Y,Z) as shown inFIG. 5. The first plane P1 includes the oversampled complex image IM3and the second plane P2 constitutes the plane for computing the hologramHO. The production of the diffracted image IM4 can be simulated in amanner that is known in the art by applying a Fourier transform to theimage IM3. In the method according to the invention, the diffractedimage IM4 is preferably determined otherwise, namely by computing, inthe plane P2, a convolutional product associated with the oversampledcomplex image IM3. This convolutional product conforms to scalardiffraction theory. For example, using a Rayleigh-Sommerfeld scalardiffraction formulation, the two components of the convolutional productcan respectively correspond to a complex field representing theoversampled complex image IM3 and a complex field representing aspherical optical wave with the same wavelength as the optical wave DIF.The skilled person however knows other types of convolutional productfor computing a diffracted image. The convolutional product computed instep E4 uses parameters including the distance D between the geometricalplanes P1 and P2 and the wavelength of the coherent optical wave DIF.

In accordance with the invention, the convolutional product is computedby applying a complex transform, also referred to as a fast complextransform, to the two components of the convolutional product, computingthe product of the resulting fast complex transforms, and then applyingthe fast complex transform which is the inverse of said fast complextransform to the aforementioned product of the fast complex transforms.

To be more precise, if CONV denotes the convolutional product, C1 and C2its two components and T the fast complex transform, then theconvolutional product is written:

CONV=C1{circle around (X)}C2=T¹T(C1{circle around (X)}C2)

CONV=T¹(T(C1)T(C2)).

In the present context, the expression “fast complex transform” means amathematical transform compatible with scalar optical diffractiontheory, i.e. whose resulting transformed functions satisfy theconventional scalar diffraction equations. The fast complex transformmust also have the property whereby the fast complex transform of aconvolutional product of two components is equal to the product of therespective fast complex transforms of each of said two components. TheFourier transform, the orthogonal polynomial transform, the Paleytransform, the Hadamar transform, the Walsh transform, the Hankeltransform, the Karhunen-Loeve transform, the multiresolution discretewavelet transform and the adaptive wavelet transform are all fastcomplex transforms which meet the above conditions. Other appropriatefast complex transforms are composites of at least two of theaforementioned transforms, such as a composite of the Walsh transformand the Hadamar transform. The application of a composite of twotransforms T1 and T2 to any image I is defined in standard mathematicalterms by the equation:

(T1·T2)(I)=T1(T2(I)).

Each of the aforementioned fast complex transforms can be used in aspecific case. In particular, the fast complex transform can be chosenaccording to the distance D between the planes P1 and P2. A Fouriertransform is appropriate for a large distance D. A Walsh transform ismore suitable for a smaller distance D. Also, it has been found thatusing one of the above-mentioned fast complex transforms other than theFourier transform gives better results in terms of the quality of thehologram HO than those obtained using the Fourier transform.

It should be noted that, because the two-dimensional image IM istransformed into a complex image IM2, computing the convolutionalproduct associated with the image IM in step E4 is more practical thanin the prior art since the fast complex transform is applied directly toan image IM3 defined by a complex function and not to an image definedby a real function.

At the exit from step E4, the diffracted image IM4 is defined by acomplex field made up of a set of complex numbers each of which isassociated with a point of the image IM4. Each of these complex numbersalso depends on the image IM3 taken as a whole.

In a next step E5 a complex field simulating a reference optical waveREF with the same wavelength as the optical wave DIF and directedtowards the hologram computation plane P2 is added, in the plane P2, tothe complex field representing the diffracted image IM4. The amplitudeinformation contained in the resulting complex field is then extractedin order to produce an interference field. The addition of theaforementioned two complex fields is performed by adding, at each pointof the diffracted image IM4, the complex number associated with thatpoint and the value at the same point of the complex field representingthe reference wave REF. The interference field constitutes the hologramHO of the two-dimensional image IM.

A variant of the FIG. 1 algorithm dispenses with the steps E1 and E2 ofproducing the complex image IM2 and/or the oversampling step E3. Inanother variant, the oversampling step E3 precedes the step E1.

The hologram HO of the two-dimensional image IM obtained in step E5 is adiffractive field, or grating, which is computed for a particularwavelength, namely the wavelength of the optical waves DIF and REF. Thishologram, which is present in virtual form in step E5, i.e. representedby digital data, is such that, if it is reproduced physically by aholographic screen, illuminating said holographic screen with a lasersource emitting at the aforementioned wavelength reproduces the originaltwo-dimensional image IM at a given order of diffraction.

The hologram HO obtained in step E5 is defined digitally by atwo-dimensional amplitude function A(u,v), where (u,v) designateco-ordinates in the hologram computation plane P2 which correspond toimage spatial frequencies when the fast complex transform chosen in stepE4 is a Fourier transform, for example. The two-dimensional amplitudefunction A(u,v) is deduced from the two-dimensional intensity functionf(x,y) describing the two-dimensional image IM, as explained above. Inpractice, the function A(u,v) is computed only for a series of discretepoints (u,v)=(u_(k), v_(q)), where k and q are integers. The values thatthe function A(u,v) takes can nevertheless be spread continuouslybetween a minimum amplitude value and a maximum amplitude value.

In a step E6 of the FIG. 1 algorithm, the values taken by the functionA(u,v) are quantized and encoded, i.e. each value of this function isassociated with a discrete value which is encoded digitally, for exampleon eight bits. To each pair of discrete points (u_(k), v_(q)) there thencorresponds a discrete amplitude value representing one of 256 graylevels. The amplitudes A(u,v) can also be quantized more simply byallocating to each amplitude value of A(u,v) the discrete value “0” ifsaid amplitude value is below a predetermined threshold or the discretevalue “1” if said amplitude value is above the predetermined threshold.

The encoding step E6 enables the hologram HO to be adapted to suitdigital display systems, such as a digitally controlled liquid crystalscreen, or to facilitate transmission of the hologram viatelecommunication systems.

What is claimed is:
 1. A method of producing a hologram from atwo-dimensional image stored in a memory and defined by a real function,said method comprising the following steps: transforming thetwo-dimensional image defined by said real function into a complextwo-dimensional image defined by a complex function, oversampling thetwo-dimensional image, simulating the production of a diffracted imageresulting from the diffraction of an optical wave by the two-dimensionalimage, and producing said hologram by adding a complex fieldrepresenting a reference optical wave to the resulting diffracted image.2. A method according to claim 1, wherein the transforming step iscarried out prior to the oversampling step.
 3. A method according toclaim 2, further comprising the step of encoding values taken by theamplitude of the sum of said complex field and the resulting diffractedimage.
 4. A method according to claim 2, wherein said transform stepcomprises the following steps: determining amplitude values eachdepending on the square root of a corresponding value taken by said realfunction, and associating a phase with each of said amplitude values sothat an amplitude value and a phase value are defined for each point ofthe complex image.
 5. A method according to claim 2, wherein saidsimulation step includes calculating a complex transform, said complextransform being a Fourier transform.
 6. A method according to claim 2,wherein said simulation step includes computing one of the followingcomplex transforms: Walsh transform, Hankel transform, orthogonalpolynomial transform, Hadamar transform, Karhunen-Loeve transform,multiresolution discrete wavelet transform, adaptive wavelet transform,a transform which is a composite of at least two of the above transformsand a transform which is a composite of at least one of the abovetransforms and a Fourier transform.
 7. A method according to claim 5,wherein said simulation step consists of computing a convolutionalproduct, associated with the oversampled complex image, of twocomponents, by applying the transform which is the inverse of saidcomplex transform to the product of the respective complex transforms ofsaid two components.
 8. A method according to claim 1, wherein thetransforming step is carried out after the oversampling step and priorto the simulating step.
 9. A system for producing a hologram from atwo-dimensional image stored in a memory and defined by a real function,characterized in that it comprises: transform means for transforming thetwo-dimensional image defined by said real function into a complextwo-dimensional image defined by a complex function, means foroversampling the complex image, simulator means for simulating theproduction of a diffracted image resulting from the diffraction of anoptical wave by the oversampled complex image, and means for producingsaid hologram by adding a complex field representing a reference opticalwave to the resulting diffracted image.
 10. A system according to claim9, further comprising means for encoding values taken by the amplitudeof the sum of said complex field and the diffracted image.
 11. A systemaccording to claim 9, wherein said transform means comprise: means fordetermining amplitude values each depending on the square root of acorresponding value taken by said real function, and means forassociating a phase with each of said amplitude values so that anamplitude value and a phase value are defined for each point of thecomplex image.
 12. A system according to claim 9, wherein said simulatormeans comprise means for computing a complex transform, said complextransform being a Fourier transform.
 13. A system according to claim 9,wherein said simulator means comprise means for computing one of thefollowing complex transforms: Walsh transform, Hankel transform,orthogonal polynomial transform, Hadamar transform, Karhunen-Loevetransform, multiresolution discrete wavelet transform, adaptive wavelettransform, a transform which is a composite of at least two of the abovetransforms and a transform which is a composite of at least one of theabove transforms and a Fourier transform.
 14. A system according toclaim 12, wherein said simulator means comprise means for computing aconvolutional product, associated with the oversampled complex image, oftwo components, by a applying the transform which is the inverse of saidcomplex transform to the product of the respective complex transforms ofsaid two components.
 15. A system for producing a hologram from atwo-dimensional image stored in a memory and defined by a real function,characterized in that it comprises: means for oversampling thetwo-dimensional image, transform means for transforming the oversampledtwo-dimensional image into a complex two-dimensional image defined by acomplex function, simulator means for simulating the production of adiffracted image resulting from the diffraction of an optical waveby theoversampled complex image, and means for producing said hologram byadding a complex field representing a reference optical wave to theresulting diffracted image.
 16. A method of producing a hologram from atwo-dimensional image which is in the form of digital data stored in amemory, the method comprising the following steps: simulating theproduction of a diffracted image resulting from the production of anoptical wave by the two-dimensional image, and producing said hologramby adding a complex field representing a reference optical wave to theresulting diffracted image, characterized in that the simulating stepconsists of computing a convolutional product, associated with thetwo-dimensional image, of two components, by applying a complextransform to the two components so as to obtain two respectivetransforms, and by further applying the transform which is the inverseof said complex transform to the product of said transforms.
 17. Amethod according to claim 16, characterized in that said complextransforms is a transform from the group consisting of a Walshtransform, a Hankel transform, a Paley transform, an orthogonalpolynomial transform, a Hadamar transform, a Karhunen-Loeve transform, amultiresolution discrete wavelet transform, an adaptive wavelettransform, a transform which is a composite of at least two of the abovetransforms, a transform which is a composite of at least one of theabove transforms, and a Fourier transform.
 18. A method according toclaim 16, characterized in that said complex transform is a Fouriertransform.
 19. A method according to claim 16, characterized in that thetwo-dimensional image is defined by a real function, and that the methodcomprises, prior to the simulating step, the step of transforming thetwo-dimensional image defined by said real function into a complextwo-dimensional image defined by a complex function.
 20. A methodaccording to claim 19, characterized in that said step of transformingthe two-dimensional image into a complex two-dimensional image comprisesthe following steps: determining amplitude values each depending on thesquare root of a corresponding value taken by said real function, andassociating a phase with each of said amplitude values so that anamplitude value and a phase value are defined for each point of thecomplex image.
 21. A method according to claim 16, characterized in thatit comprises, prior to the simulating step, the step of oversampling thetwo-dimensional image.
 22. A method according to claim 21, in that theoversampling step is carried out after said step of transforming thetwo-dimensional image into a complex two-dimensional image.
 23. A methodaccording to claim 16, further comprising the step of encoding valuestaken by the amplitude of the sum of said complex field and theresulting diffracted image.
 24. A method according to claim 16,characterized in that the two components of the convolutional productrespectively correspond to a complex field representing thetwo-dimensional image and to a complex field representing a sphericaloptical wave.
 25. A system for producing a hologram from atwo-dimensional image stored in a memory, comprising: means forsimulating the production of a diffracted image resulting from theproduction of an optical wave by the two-dimensional image, and meansfor producing said hologram by adding a complex field representing areference optical wave to the resulting diffracted image, characterizedin that the simulating means comprise means for calculating aconvolutional product, associated with the two-dimensional image, of twocomponents, by applying a complex transform to the two components so asto obtain two respective transforms, and by further applying thetransform which is the inverse of said complex transform to the productof said transforms.
 26. A system according to claim 25, characterized inthat said complex transforms is a transform from the group consisting ofa Walsh transform, a Hankel transform, a Paley transform, an orthogonalpolynomial transform, a Hadamar transform, a Karhunen-Loeve transform, amultiresolution discrete wavelet transform, an adaptive wavelettransform, a transform which is a composite of at least two of the abovetransforms, a transform which is a composite of at least one of theabove transforms, and a Fourier transform.
 27. A system for according toclaim 25, characterized in that said complex transform is a Fouriertransform.
 28. A system for according to claim 25, characterized in thatit further comprises means for transforming the two-dimensional imagedefined by a real function, into a complex two-dimensional image definedby a complex function.
 29. A system for according to claim 28,characterized in that said means for transforming the two-dimensionalimage into a complex two-dimensional image comprises: means fordetermining amplitude values each depending on the square root of acorresponding value taken by said real function, and means forassociating a phase with each of said amplitude values so that anamplitude value and a phase value are defined for each point of thecomplex image.
 30. A system for according to claim 25, characterized inthat it further comprises means for oversampling the two-dimensionalimage.
 31. A system for according to claim 25, characterized in that itfurther comprises means for encoding values taken by the amplitude ofthe sum of said complex field and the resulting diffracted image.
 32. Asystem for according to claim 25, characterized in that the twocomponents of the convolutional product respectively correspond to acomplex field representing the two-dimensional image and to a complexfield representing a spherical optical wave.